For the Salted Olive Oil Dulce De Leche: In a large saucepan whisk together the milk, sugar, vanilla bean seeds + pod and baking soda and bring to a boil. Reduce to a simmer and set a timer for 1 hour. Whenever you walk by the pot, give it a stir and adjust the temperature to keep it from boiling. When the timer goes off, check the color...has it started to change a wee bit? Usually between the 1 to 1 1/2 hours mark, the color starts to deepen and the mixture begins to thicken a bit. When this happens, it's going to be more prone to boiling (which is bad), so stick near it adjusting the temperature as necessary and stirring more frequently. It should be done anywhere between 1 1/2 to 2 hours. You want the dulce to have a nice thick consistency as well as a bronzed and caramel appearance. Don't raise the temperature if it's not there yet, exercise patience (grab a glass of wine?) and continue to stir and meditate.
Once the mixture has reduced and the color is deep and rich, strain it into a heat-safe bowl. Don't skip straining!!! You want a nice, silky texture and this helps you towards that goal. Slowly stream in the olive oil, whisking the entire time to completely incorporate the oil into the dulce de leche. Sprinkle with the sea salt and whisk for one minute more. Pour into a lidded container and stash in the fridge until ready to use. The mixture will thicken up a wee more as it cools.
For the Cookies: Add the butter and powdered sugar to an electric stand mixer fitted with the paddle and mix on medium for about 4 minutes until smooth. Be sure to scrape down the sides and bottom of the bowl to make sure everything is incorporated. With the machine on low, add the eggs one at a time making sure the first is well blended before adding in the last. Add in the milk and vanilla and run the machine for 1 minute more. The mixture may appeared a bit curdled, this is fine.
In a medium bowl whisk together the flour, corn starch and sea salt. Add to the mixer and run the machine for 1 minute on low or until the mixture just comes together. On a floured surface and with floured hands, pat the dough into a disc. This is a damp dough so don't be super shy with flour as needed. Wrap the dough tightly and stash in the fridge for at least 1-2 hours or up to overnight.
Cover several baking sheets with parchment paper. Set aside.
Roll out the dough between two floured pieces of parchment paper. If you have a silpat, place it beneath the parchment paper to keep it from sliding around on the counter. Sprinkle the dough with additional flour to keep it from sticking and roll the dough 1/4 to 1/2 inch thick. Don't go too thin. Place the dough in the freezer for 10-15 minutes (or the fridge for 20 minutes) to firm up. Use a 2-inch cookie cutter and cut as many circles as possible and place on the prepared baking sheets. Place the cookie cutouts in the freezer for 10-15 minutes (or the fridge for 20 minutes).
Preheat your oven to 350F and place a rack in the top third of your oven, at least 6-inches from the heat source.
Bake the cookies in the top third of the oven for 9 minutes. Let cool on the sheet for 10 minutes and then transfer to a rack to finish cooling.
To Assemble: Cover a baking sheet with parchment paper and have a cooling rack nearby.
Add the chocolate and shortening to a medium heat-safe bowl set over a saucepan of simmering water over low heat. Stir frequently. Once melted set on top of a towel on the counter and stir frequently to help it cool evenly. Continue to stir periodically while you prep the cookies. If the chocolate starts to become less liquid, set the bowl back on the saucepan with the heat turned off.
Put a nice dollop of dulce de leche on the flat side of a cookie and top with a right-side up cookie. Set them on the prepared baking sheet and place in the freezer for 1 hour. Once firm and chilled through, dip them in the cooled chocolate, using two forks to help you dip into the chocolate and set on the cooling rack. Sprinkle with sea salt flakes if you're feeling it. Once all of the cookies are dipped in chocolate (they should start to set immediately), use a spatula to gently get them off of the rack and set them back on the baking sheet to set completely. They are ready to serve immediately. I really like keeping them in the fridge and then bringing them out a half hour-ish prior to serving.